Comparison of Strehl Ratio, Marechal Approximation, and Power in a Bucket Calculations

Transcription

1 AN023 Comparison of Strehl Ratio, Marechal Approximation, and Power in a Bucket Calculations Author: Justin Mansell Revision: 4/1/17 Abstract This application note explores the relationship between Strehl Ratio (SR), the Marechal approximation, and power in a bucket (PIB) metrics for a circular beam with select Zernike and sinusoidal aberrations. It also looks at this relationship for Kolmogorov spectrum turbulence for different bucket sizes. Model Setup This model was run in Matlab using some existing functions for things like Zernike polynomials, farfield propagation calculations, creating grids, and plotting. The appendix shows the code for the model with sinusoidal gratings, but does not include the support scripts, although those should be easily duplicated. We used two separate scripts for analysis of Zernikes and sinusoidal phase gratings. In each of the scripts, the outer most loop scans over different aberration amplitudes. Each aberration was scaled to make the root-mean-square (RMS) amplitude the same for different aberration shapes. The next inner loop scaled over aberration shape. In each shape-amplitude combination, the aberration is applied to a circular beam on a mesh with a factor of two guard-band (dxy = 2 * Dap / nxy). The Strehl ratio is calculated using the integral form of the equation. The farfield intensity of the aberrated beam is determined and then the power in the bucket calculation is performed. The power in the bucket to total power ratio is then stored for a 1x, 1.5x and 2x diffraction limited bucket size. The Marechal approximation was calculated using, StrehlMarechal where φrms is the RMS aberration amplitude. exp In each of these studies, the wavelength was 1 μm, the circular top-hat beam was 1 cm in diameter, the mesh was sampled with 256 points, and the beam was propagated 25 m to the focus. 2 RMS 1

2 Results with Zernike Polynomial Aberrations We studied Zernikes from the second order (#3) to the 33 rd Zernike (n-l ordering) and then also Zernike #100. Figure 1 shows the results for the focus aberration. Figure 2 shows the results for the 100 th Zernike, which is a 5 th -radial-order polynomial with 10 circular segments. In general, for all of the Zernikes, we found Strehl ratio and Marechal approximation matched well up to about 0.15 waves RMS amplitude. The power in the diffraction limited bucket matched reasonably well with the Strehl ratio and Marechal approximation up to this point as well. In general, the agreement between all of these metrics increased as the aberration spatial frequency increased, as is evidenced by the good agreement in Figure 2. Figure 1 - Results for a Focus Aberration Figure 2 - Results for the 100 th Zernike 2

3 Results with Sinusoidal Phase Grating Aberrations In the case of the sinusoidal gratings, we studied gratings with spatial periods equal to the aperture diameter divided by a factor between 1 and 10 in unit increments. Figure 3 and Figure 4 show the results for the lowest and highest spatial frequencies we studied respectively. The results, in general, were consistent with the conclusions of the Zernike polynomial study. The low spatial frequency aberrations were the most different than the Strehl ratio and Marechal approximation results, but the power in the diffraction limited bucket curve seemed to match reasonably well with the Strehl results. Figure 3 - Results with the lowest spatial frequency grating Figure 4 - Results with the highest spatial frequency grating For the sinusoidal aberrations, we also looked at the difference between the PIB and the Strehl ratio calculations. Figure 5 shows the result for the highest spatial frequency. There is an initial difference due to the light diffracted beyond the bucket even with no aberration amplitude, but this only decreases as the aberration amplitude increases. The maximum value of this is about 16% for the 1x diffraction limited bucket size. 3

4 Figure 5 - Difference between PIB and Strehl Ratio for the highest spatial frequency aberration PIB and Strehl Ratio for Kolmogorov Spectrum Turbulence We performed a similar calculation to those above for a 20 th -order super-gaussian beam with varying amplitude Kolmogorov-spectrum turbulence. Figure 6 shows the results averaged over 100 random realizations. The relationship between the power in the diffraction-limited spot diameter and the Strehl ratio is nearly linear, but has more structure as the bucket size is increased to 2 times and 3 times the diffraction limited spot size. Conclusions Figure 6: PIB vs. Strehl Ratio for Varying Amplitude Kolmogorov Turbulence In this application note, we have shown the great deal of similarity between the Strehl ratio, the Marechal approximation, and the power in a diffraction limited bucket. We extended this study to examine the relationship between power in the bucket and Strehl ratio when a nearly top-hat beam is aberrated with a single Kolmogorov-spectrum turbulence phase screen. 4

5 Appendix: Comparison Code for Sinusoids setup; dbg=0; ppt=1; nxy = 512; Dap = 1e-2; wavelength = 1.0e-6; dxy = Dap*2/nxy; rzern = Dap/2; SF = 5.0; %far field scaling %scanning parameters rbucket = [ ]; zvec = ([1:1:10]); ampsfv = [0 0.01:0.02:0.5]; if (ppt) TitleSlidePowerPoint('Study of comparison of Strehl, Marechal, and Power in a Bucket'); xxx=whos; for ii=1:length(xxx); eval(sprintf('val=%s;',xxx(ii).name)); txt{ii} = sprintf('%s = %f',xxx(ii).name,val); TextToPowerPoint(txt,'Variables',12,16); clear xxx; clear txt; clear val; TextFileToPPT('MarechalStrehlPIBStudy_Sinusoids.m'); %calcualte ideals g=makegrid(nxy,dxy); E0 = aperture(nxy,dxy,dap/2); [Effi,dxout,xout,zff]=FarField(E0,dxy,wavelength,SF); rdl = 1.22 * zff * wavelength / Dap; [rv,ps,pseg]=powerinbucket(abs(effi).^2,dxout); PbIdeal = Ps./ max(ps); %RMS aperture ap = aperture(nxy,dxy,rzern,0,0,0,1); b=0; for ampsf=ampsfv; nf([ ]); b=b+1; rmsamp = wavelength * ampsf; c = 0; for ii=zvec; c=c+1; 5

6 Kx = 2.0*pi/(Dap/ii); z = sin(g.xx.* Kx).* sin(g.yy.*kx); %nf; show(z); return; %scale RMS z = z.* rmsamp./(nonzerorms(z.*ap)); E = E0.* exp(1j.*(2.0*pi/wavelength).*z); SR(b,c) = Strehl(E); [Eff,dxout,xout,zff]=FarField(E,dxy,wavelength,SF); rdl = 1.22 * zff * wavelength / Dap; %[P,Ps,x]=PIB(Eff,dxy); [rv,ps]=powerinbucket(abs(eff).^2,dxout); Pb = Ps./ max(ps); rb = rv./rdl; index(c)=ii; for jj=1:length(rbucket); [mv,mi(jj)]=min(abs(rb - rbucket(jj))); Pbs(jj,b,c) = Pb(mi(jj)); if (dbg) clf; subplot(2,2,1); show(abs(e)); drawnow; title(ii); subplot(2,2,2); show(z); drawnow; title('phase'); subplot(2,2,3); show(abs(eff)); drawnow; title('far-field Field'); subplot(2,2,4); plot(rb,pb); plot(rb,pbideal,'k--'); ax=axis; ax(2)=5; axis(ax); drawnow; title('pib'); pause(0.001); close gcf; nf; clear leg; for jj=1:size(pb,1); Pbv = Pbs(1,b,:); Pbv = Pbv(:); plot(index,pbv,getlinespec(jj)); leg{jj}=sprintf('r=%.1f',rbucket(jj)); plot(index,sr(b,:),'g^-'); leg{end+1}=sprintf('strehl'); plot(index,index.*0 + exp(-1* ((ampsf*2*pi).^2))) leg{end+1}=sprintf('marechal'); ax=axis; ax(3)=0; axis(ax); legend(leg); xlabel('zernike'); ylabel('value'); 6

7 %% display final results nf; plot(ampsfv,exp(-1*((ampsfv*2*pi).^2)),'k--*','linewidth',2); for ii=1:3; %size(pbs,3); Pbv = Pbs(1,:,ii); Pbv = Pbv(:); plot(ampsfv,pbv,getlinespec(ii)); plot(ampsfv,sr(:,ii),getlinespec(ii,'colorandmarker')); plot(ampsfv,exp(-1*((ampsfv*2*pi).^2)),'k--*','linewidth',2); xlabel('rms Amplitude (waves)'); ylabel('ampliutde'); legend('marechal','p Ratio','Strehl'); %% results by aberration shape nf([ ]); for ii=1:length(zvec); clf; subplot(1,2,1); Kx = 2.0*pi/(Dap/zVec(ii)); z = sin(g.xx.* Kx).* sin(g.yy.*kx); show(z.*ap); title(sprintf('cycles Per Aperture: %i',zvec(ii))); subplot(1,2,2); clear leg; for jj=1:length(rbucket); Pbv = Pbs(jj,:,ii); Pbv = Pbv(:); plot(ampsfv,pbv,getlinespec(jj)); leg{jj} = sprintf('r_b_u_c_k_e_t=%.1f',rbucket(jj)); plot(ampsfv,sr(:,ii),getlinespec(length(rbucket)+1)); leg{end+1}=sprintf('strehl Ratio'); plot(ampsfv,exp(-1*((ampsfv*2*pi).^2)),'k--','linewidth',2); leg{end+1}=sprintf('marechal Approx.'); legend(leg,'location','best'); drawnow; if (ppt) ToPPT(); 7